796 research outputs found

    Independence-friendly cylindric set algebras

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    Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate.Comment: 42 pages. Submitted to the Logic Journal of the IGPL. See also http://math.colgate.edu/~amann

    Lattice initial segments of the hyperdegrees

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    We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, Dh\mathcal{D}_{h}. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of Dh\mathcal{D}_{h}. Corollaries include the decidability of the two quantifier theory of % \mathcal{D}_{h} and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω1CK\omega _{1}^{CK}. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω1\omega _{1}. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of Dh\mathcal{D}_{h}

    On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos

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    In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs

    Scopes and Limits of Modality in Quantum Mechanics

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    We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.Comment: 9 pages, no figure

    Free QQ-distributive lattice over an nn-element chain

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    In this note we provide an explicit construction of FQ(n)FQ({\bf n}), the free QQ-distributive lattice over an nn-element chain, different from those given by Cignoli [4] and Abad--Díaz Varela [1], and prove that FQ(n)FQ({\bf n}) can be endowed with a structure of a De Morgan algebra.Fil: Monteiro, Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Abad, Manuel. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Zander, Marta Amalia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentin
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